Problem 44 - University Physics with Modern Physics

Question

An astronaut is tethered by a strong cable to a spacecraft. The astronaut and her spacesuit have a total mass of 105 \(\mathrm{kg}\) , while the mass of the cable is negligible. The mass of the spacecraft is \(9.05 \times 10^{4} \mathrm{kg}\) . The spacecraft is from any large astronomical bodies, so we can ignore the gravitational forces on it and the astronaut. We also assume that both the spacecraft and the astronaut are initially at rest in an inertial reference frame. The astronaut then pulls on the cable with a force of 80.0 \(\mathrm{N}\) . (a) What force bodies, so we can ignore the gravitational forces on it and the astronaut. We also assume that both the spacecraft and the astronaut are initially at rest in an inertial reference frame. The astronaut then pulls on the cable with a force of 80.0 \(\mathrm{N}\) . (a) What force does the cable exert on the astronaut? (b) Since \(\Sigma \overrightarrow{\boldsymbol{F}}=m \overrightarrow{\mathbf{a}},\) how can a "massless" \((m=0)\) cable exert a force? (c) What is the astronaut's acceleration? (d) What force does the cable exert on the spacecraft? (e) What is the acceleration of the spacecraft?

Step-by-Step Solution

Answer

(a) 80.0 N; (b) Cable transmits tension; (c) 0.762 m/s²; (d) 80.0 N; (e) 8.84 × 10⁻⁴ m/s².

Step 1: Analyze Forces on the Astronaut

According to Newton's third law, the force that the cable exerts on the astronaut is equal in magnitude and opposite in direction to the force the astronaut exerts on the cable. Therefore, the force exerted by the cable on the astronaut is also: \[ F = 80.0 \, \mathrm{N} \]

Step 2: Understand Force Exertion by Cable

Though the cable is assumed to have negligible mass, it can still transmit forces due to tension. The tension is created by the force exerted by the astronaut, and acts along the length of the cable. Hence, the cable transmits this tension force to the spacecraft.

Step 3: Calculate Astronaut's Acceleration

Use Newton's second law, \( \Sigma \overrightarrow{F} = m \overrightarrow{a} \), to find the astronaut's acceleration. The net force on the astronaut is the force exerted by the cable. \[ a_{\text{astronaut}} = \frac{F}{m_{\text{astronaut}}} = \frac{80.0}{105} \, \mathrm{m/s^2} \approx 0.762 \, \mathrm{m/s^2} \]

Step 4: Determine Force on the Spacecraft

The force exerted by the cable on the spacecraft is equal in magnitude to the force exerted by the astronaut on the cable, as per Newton's third law. Thus, the force is:\[ F = 80.0 \, \mathrm{N} \]

Step 5: Calculate Spacecraft's Acceleration

Use Newton's second law for the spacecraft. The force exerted on it is the force transmitted by the cable.\[ a_{\text{spacecraft}} = \frac{F}{m_{\text{spacecraft}}} = \frac{80.0}{9.05 \times 10^4} \, \mathrm{m/s^2} \approx 8.84 \times 10^{-4} \, \mathrm{m/s^2} \]

Key Concepts

Tension Force

In physics, a tension force is the pulling force transmitted along a flexible object, like a string, rope, or cable. Tension is always directed along the length of the cable and pulls equally on the objects it is attached to on either end. In this exercise, the astronaut applies an 80.0 N force on the cable. Consequently, due to Newton's Third Law, the cable applies an equal and opposite tension force of 80.0 N back on the astronaut.

Important characteristics of tension include:

• The force acts along the length of the cable.
• Tension is uniform throughout a cable if it is assumed massless.
• The cable transmits forces without alteration if it’s massless.

Tension force plays a crucial role in scenarios where one object pulls on another, helping us analyze the force interactions and motion of different bodies.

Massless Object

In many physics problems, certain objects, such as cables or ropes, are considered massless to simplify calculations. A massless cable means its mass is negligible compared to other masses involved in the system. In our exercise, the cable is considered massless, which means:

• The cable will not have any inertia or mass-related effects.
• It allows the tension throughout the cable to be uniform.
• The cable transmits forces applied to it without any loss or attenuation.

This assumption is crucial when making calculations related to tension as it implies that the cable only serves as a means to transfer forces between two or more objects, with no impact of its own due to gravity or inertia.

Acceleration Calculation

Acceleration is calculated using Newton's Second Law, which states that the sum of forces acting on an object equals the mass of that object times its acceleration, given by the equation \( \Sigma \overrightarrow{F} = m \overrightarrow{a} \).

In the exercise, we calculate the astronaut's acceleration by applying the force formula:

• The net force exerted is 80.0 N (tension force from the cable).
• The total mass of the astronaut and suit is 105 kg.
• Acceleration of the astronaut: \( a_{\text{astronaut}} = \frac{80.0}{105} \approx 0.762 \text{ m/s}^2 \).

This calculation helps us understand how effectively the force produces motion in the astronaut, demonstrating the impact of force magnitude and mass on acceleration.

Inertial Reference Frame

An inertial reference frame is a perspective in which Newton’s laws of motion are valid. In other words, objects at rest remain at rest, and objects in motion continue in straight lines at constant speeds unless acted upon by external forces. In the given exercise, the spacecraft and astronaut are initially at rest in an inertial reference frame, implying:

• No initial forces were acting on the system.
• The motions considered arise due to forces applied and not due to any inherent non-inertial effects.
• Any accelerations calculated are due only to the applied forces like tension.

Using an inertial reference frame simplifies the analysis of forces and motion, making it easier to apply and understand Newton's laws to solve problems related to motion.